Suppose you are given two indexed permutations, (7 followed by 4, for instance)

1. What is the best way to go about finding their composition, given the indices themselves? I'd imagine that the answer would involve converting the indices to some cycle form, performing the necessary computation then finding the index of the resulting permutation. Does there exist a more concise method to compose two arbitrary permutations, or does the index not inherently contain enough information?
2. Can this information be used to study permutation groups (like $S_{4}$ or $M_{24}$ etc.), or is the index missing important details? Like the permutation's conjugacy class?
3. Can this operation be extended to apply to any two whole numbers? i.e. Can there exist an infinite group of permutations, applicable as a noncommutative binary operation over $\mathbb N_{0}$?